Hypergraphs with irrational Tur\'{a}n density and many extremal configurations

TL;DR

This paper constructs a 3-uniform hypergraph family with irrational Turán density and multiple distinct near-extremal configurations, revealing new complexities in hypergraph extremal problems and the structure of feasible regions.

Contribution

It introduces the first example of a hypergraph family with irrational Turán density and multiple far-apart near-extremal configurations, highlighting non-stability in hypergraph extremal theory.

Findings

Turán density of the constructed hypergraph family is irrational.

Existence of multiple, structurally diverse near-extremal configurations.

First example demonstrating irrational Turán density with non-stability in hypergraphs.

Abstract

Unlike graphs, determining Tur\'{a}n densities of hypergraphs is known to be notoriously hard in general. The essential reason is that for many classical families of $r$-uniform hypergraphs $\mathcal{F}$, there are perhaps many near-extremal $\mathcal{M}_t$-free configurations with very different structure. Such a phenomenon is called not stable, and Liu and Mubayi gave a first not stable example. Another perhaps reason is that little is known about the set consisting of all possible Tur\'{a}n densities which has cardinality of the continuum. Let $t\ge 2$ be an integer. In this paper, we construct a finite family $\mathcal{M}_t$ of 3-uniform hypergraphs such that the Tur\'{a}n density of $\mathcal{M}_t$ is irrational, and there are $t$ near-extremal $\mathcal{M}_t$-free configurations that are far from each other in edit-distance. This is the first not stable example that has an irrational Tur\'{a}n density. It also provides a new phenomenon about feasible region functions.